Parametric Solitons in isotropic mediaParametric Solitons in isotropic media

D. A. Georgieva, L. M. Kovachev

Fifth Conference AMITaNS June 24 - 29, 2013, Albena, Bulgaria 

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The effects of confinement between picosecond optical pulses due to Cross Phase Modulation (CPM) in multimode and birefringent optical fibres have been discussed in many papers 1) Menyuk (1989, 2004); 2) Afanas’ev, Kovachev and Serkin (1990); 3) Kovachev (1991); 4) Radhakrishnan, Lakshmanan and Hietarinta (1997); 5) Todorov and Christov (2009).

The basic effect in the collision dynamics between two or three pulses is a detachment and confinement of part of one pulse in the other. The obtained mixed states are stable at long distances.

The numerical experiment shows that during the propagation and interaction the localized waves change their shape, but they preserve the quantities normally associated with the coupled Nonlinear Schrëdinger Equations (NLSs). In a stable mixed state each of the pulses conserves his mass (local energy).

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-4 0 4

The typical evolution of optical pulses governed by the coupled Nonlinear Schrëdinger Equations.

For the numerical experiments we use the Split-Step Fourier Method.

4

Is it possible a mechanism of an exchange of energy

between the pieces into the mixed states?

We will discuss this possibility on the base of two types of

parametric processes in optical fibers.

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NONLINEAR POLARIZATION OF TWO COMPONENTS AT ONE CARRYING FREQUENCY

The electrical field associated with linearly or elliptically polarized optical wave is

,..exp2

1,,, 0 cctiAyAx

itzyxE yx

where Ax, Ay are the complex amplitudes of the polarization components of a wave with a carrying frequency ω0.

6

We investigate the polarization dynamics of two initially linearly polarized components of the electrical field

1) there is no initial phase difference between the components; 2) the complex amplitudes can be expressed as a product of two amplitude functions with equal phases: Ax = Cxexp(i), Ay = Cyexp(i).

The nonlinear polarization in the case of isotropic medium is well known

LINEARLY POLARIZED COMPONENTS. MANAKOV SYSTEM

,2

** EEEB

EEEAPnl

where ω is the optical frequency, A(ω), B(ω) are nonlinear coefficients. The components of the nonlinear polarization are

,2

22

xyxx

nl AAAB

AP

.2

22

yyxy

nl AAAB

AP

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Manakov system

From Maxwell equations in the negative dispersion region of an optical fibre

where is a dimensionless nonlinear parameter. (The system is written in normalized by dispersion lengths z = z/zdisp and moving with the group velocity frame (Local Time frame): z = z, t = t - z/v.)

The Manakov system admits the following fundamental stable soliton solution when = 1

,02

1 22

2

2

xyxxx AAA

t

A

z

Ai

.02

1 22

2

2

yyx

yy AAAt

A

z

Ai

.2

expsec2

1,

2expsec

2

1

izthA

izthA yx

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The exact solution is invariant with respect to an arbitrary initial phase difference, i.e.

also is a fundamental soliton solution of the Manakov system.

21 2

expsec2

1,

2expsec

2

1 iiz

thAiiz

thA yx

ELLIPTICALLY POLARIZED COMPONENTS.

In the case of elliptically polarized light pulses the complex amplitudes Ax and Ay as a product of two amplitude functions with different initial phases: Ax = Cxexp(ix), Ay = Cyexp(iy).

In the case of nonresonant electronic nonlinearities A(ω) = B(ω) = const and

,3

1

3

2 *22

yyxxyx

xnl AAAAAAconstP

.3

1

3

2 *22

xxyyxy

ynl AAAAAAconstP

Last terms present degenerate four wave mixing process (ω1 = ω2 = ω3).

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Parametric system

From Maxwell equations in an optical fibre

In this case x y 0 (phase difference) periodical exchange of energy between the elliptically polarized components.

When the phase difference vanishes (x y = 0 – linearly polarized components) 1) the amplitude functions admit again equal phases;2) the nonlinear polarization with elliptically polarized components can be transfor- med to the nonlinear orthogonal polarization (Manakov case);3) the parametric terms appear as usual cross-phase terms and the Parametric

system is equal to the Manakov system.

,03

1

3

2

2

1 *22

2

2

yyxxyxxx AAAAAA

t

A

z

Ai

.03

1

3

2

2

1 *22

2

2

xxyyxy

yy AAAAAAt

A

z

Ai

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Exact soliton solution

The Parametric system has the same fundamental soliton solution as the Manakov system when = 1 and if there is not an initial phase difference ( = 1 - 2 = 0)

but it is not invariant with respect to an arbitrary initial phase difference, i.e.

,2

expsec2

1,

2expsec

2

1

izthA

izthA yx

21 2

expsec2

1,

2expsec

2

1 iiz

thAiiz

thA yx

is not a fundamental solution of the Parametric system.

Overlapped components

When we investigate initial components with phase difference = 1 - 2 0 and the initial amplitudes are slightly above the theoretical for soliton regime we observe a stable propagation with a periodical exchange of energy between the components.

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= π / 2

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Initially separated components. Collision dinamics

When the components of the pulse 1) are initially spatially separated; 2) have different phases; and 3) the degenerated condition of the wave synchronism is satisfied (2ω1 - 2ω2 = 0) the components collide. After the collision we observe a detachment and confinement of part of one pulse in the other. We observe a generation of mixed states of the wave packets and an intensive energy exchange during their propagation.

t -4 0 4

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HAMILTONIAN STRUCTURE OF THE MANAKOV TYPE SYSTEMS AND THE PARAMETRIC SYSTEM

These systems 1) are conservative; 2) have Hamiltonian structure; 3) satisfy at least three conservation laws of a) Hamiltonian H, b) momen- tum P and c) total energy N.

MANAKOV TYPE SYSTEMS

We consider

,02

1 22

2

2

xyxxx AAaA

t

A

z

Ai

,02

1 22

2

2

yxy

yy AAaAt

A

z

Ai

where a > 0.

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Let us consider the following remaining integrals

a) the total energy of the system

b) the energy of each component – at every moment of the interaction, even in the case when the components could be separated on different pieces, the localized energy is constant

.2

1 2244**

yxyxyyxx AAaAA

t

A

t

A

t

A

t

AH

The Hamiltonian of the Manakov type systems:

,22

21 constdtAdtANNN yx

.

,

2

2

2

1

constdtAN

constdtAN

y

x

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PARAMETRIC SYSTEM

.3

1

3

2

2

1 2*222*2244**

yxyxyxyxyyxx AAAAAAAA

t

A

t

A

t

A

t

AH

The Hamiltonian of the Parametric system has the form

a) the total energy of the system is still constant

,22

21 constdtAdtANNN yx

b) the energy of each component is not a constant

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the complex amplitudes Ax and Ay are presented in exponential form – .exp,exp yyyxxx iAAiAA

The change of energy of each pulse is described by the first derivative of local energy

,22sin3

2

3

1 222*222*1 dtAAdtAAAAiz

Nyxyxyxyx

.22sin3

2

3

1 2222*2*22 dtAAdtAAAAiz

Nyxyxyxyx

• When = 2x 2y = 0 (Manakov case) the energy of each component is a constant and there is not an exchange of energy.

• When = 2x 2y 0 we observe an exchange of energy between the waves which reaches its maximum when = 2x 2y = π / 2.

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DIFFERENT CARRYING FREQUENCIES, CROSS-PHASE MODULATIONAND PARAMETRIC AMPLIFICATION

Using optical fibres as a nonlinear medium, a wide variety of nonlinear effects have been observed. One of them, the four photon parametric mixing between three laser pulses (2ω3 = ω1 +ω2) can be used to convert the input light pulse at ω3 frequency on two different frequencies ω1 and ω2.

The electrical field associated with three linear polarized optical waves at different frequencies can be written in the form

where A1, A2 and A3 are the complex amplitudes of the components of three waves on different frequencies and the phase-matching condition 2ω3 =ω1+ω2 is satisfied.

cctiAtiAtiAi

xtzyxE .expexpexp2

1,,, 332211

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Parametric system

The corresponding Parametric system of the amplitude equations, governing the nonlinear propagation in a fibre, written in approximation up to second order of dispersion is

where v is the group velocity, βi are the dispersion parameters and is the nonlinear coefficient.

,0exp222

1 *2

231

2

3

2

2

2

121

2111

zkiAAAAAAt

A

t

A

vz

Ai

,0exp222

1 *1

232

2

3

2

2

2

222

2222

zkiAAAAAAt

A

t

A

vz

Ai

,0exp2222

*3213

2

2

2

1

2

323

233

zkiAAAAAAAt

A

z

Ai

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Exact soliton solution

The Parametric Nonlinear System of Equations (PNSE) admits an exact soliton solution of the kind

when the waves have equal group velocities, kz = 0 and β1 = β2 = β3 = 1. These parametric solitons are with equal initial phases and are relatively stable on large distances.

Overlapped puses

Let us investigate PNSE for initial pulses with phase difference – when there is an initial phase difference Δϕ = π / 2 and the initial amplitudes are slightly above the theoretical for soliton regime, for example

,2

expsec5

1,

2expsec

15

2,

2expsec

15

2321

izthA

izthA

izthA

,sec5

1,

4expsec

5

1321 thA

ithAA

we observe an intensive exchange of the energy between the waves.

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a) the exact soliton solution b) overlapped pulses

thA

thAA

sec5

1

;sec15

2

3

21

thA

ithAA

sec5

1

;4

expsec15

1

3

21

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Initially separated pulses. Collision dinamics

When the pulses 1) are initially separated; 2) have different phases; 3) have different group velocities and 4) the condition of the wave synchronism is satisfied (2ω3 = ω1 + ω2) the pulses collide. We observe a generation of mixed states of the wave packets and an intensive energy exchange between them.

t -6 0 6

4/1 )6(sec iethA

)(sec3 thA 4/

2 )6(sec iethA

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1) We investigate the propagation of two and three optical pulses in an isotropic media when

phase-matched conditions of the kind (2ω1 −2ω2 = 0) or (2ω3 = ω1 +ω2) are satisfied.

2) In this case terms connected with four-photon parametric processes in the corresponding

nonlinear evolution equations appear. These terms generate a periodical exchange of the

energy between the optical waves.

3) The existence of new kind of parametrically connected solitons is discussed. We observe a

confinement of the waves and a generation of mixed states of wave packets with different

polarization or frequencies.

4) It is shown that the Parametric system of equations has Hamiltonian with structure quite

different than the Hamiltonians of the well known Manakov type systems.